Another Brit here. I did a west to east Southern Tier in 2013. Started mid March. Temps were not a problem anywhere. A few hot days in the first week. Start early drink plenty fluids. Below freezing overnight in Silver City, New Mexico.
March/April is an ideal time for the Southern Tier.
I did a west to east ride in 2001. I had snow in mountains above San Diego on March 9th which was abnormal. Also below freezing near the AZ/NM border. However most ideal temperatures everywhere.
I did a ride between Tucson and El Paso in December 2019 and had snow overnight once but generally cool but rideable temperatures.
I also did a ride between El Paso and Austin in May 2018. The day I left Langtry, May 8th, was officially 95F and my bike computer registered 102F on hot pavement. Much warmer than that and I'll not be out for extended periods riding.